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a) Let $A:=\left(\begin{array}{ccc}1 & \frac{1}{2} & 0 \\ \frac{1}{2} & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$. Briefly show that the bilinear map $\mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R}$ defined by $(x, y) \mapsto x^{T} A y$ gives a scalar product.
b) Let $\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be the linear functional $\alpha:\left(x_{1}, x_{2}, x_{3}\right) \mapsto x_{1}+x_{2}$ and let $v_{1}:=(-1,1,1), v_{2}:=(2,-2,0)$ and $v_{3}:=(1,0,0)$ be a basis of $\mathbb{R}^{3}$. Using the scalar product of the previous part, find an orthonormal basis $\left\{e_{1}, e_{2}, e_{3}\right\}$ of $\mathbb{R}^{3}$ with $e_{1} \in \operatorname{span}\left\{v_{1}\right\}$ and $e_{2} \in \operatorname{ker} \alpha$.
151. Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map defined by the matrix $A$. If the matrix $B$ satisfies the relation $\langle A X, Y\rangle=\langle X, B Y\rangle$ for all vectors $X \in \mathbb{R}^{n}, Y \in \mathbb{R}^{k}$, show that $B$ is the transpose of $A$, so $B=A^{T}$. [This basic property of the transpose,
$\langle A X, Y\rangle=\left\langle X, A^{T} Y\right\rangle$
is the only reason the transpose is important.]
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